VaR Estimation under Stochastic Volatility Models
|
|
- Augustus Anthony
- 5 years ago
- Views:
Transcription
1 VaR Estimation under Stochastic Volatility Models Chuan-Hsiang Han Dept. of Quantitative Finance Natl. Tsing-Hua University TMS Meeting, Chia-Yi (Joint work with Wei-Han Liu) December 5, 2009
2 Outline Risk Management in Practice: Value at Risk (VaR) Estimate Default Probability by Efficient Importance Sampling Fourier Transform Method: boundary effect and a price correction scheme Stability of Estimation and some Empirical Results 1
3 Value at Risk Let r(t) be an asset return at time t. Its (1 α)% VaR, denoted by V ar α, is defined by the α%-quantile of r(t). That is, P (r(t) VaR α ) = 1 α. That is a risk controller has a (1 α)% confidence that the asset price will not drop below V ar α in time t. 2
4 Aspects about VaR Mathematically, it is not a coherent risk measure because it doesn t satisfy the risk diversification principal. Instead, CVaR is. Practically, it is commonly required by financial regulations. Artzner P., F. Delbaen, J.-M. Eber, and D. Heath, Coherent Measures of Risk, Mathematical Finance, 9 (1999):
5 Estimation of VaR Riskmetrics: normal assumption Historical Simulation: generate scenarios Monte Carlo method: model dependent 4
6 Estimate Probability of Default Given a dynamical model of an asset price S t, its return process is r t = ln S t /S t 1. Given a loss threshold B, the probability of default is defined by DP(B) = E t 1 {I(r(t) B)}. Note: V ar α is the B satisfying DP (B) = α. 5
7 Importance Sampling Given the Black Scholes Model under measure P, a new measure P defined from an exponential martingale dp d P = Q satisfies Ẽ t 1 [S t ] = exp( B). Denote DP by P ε and the second moment by M 2ε, which are defined by P ε = IE t 1 [I (r t B)] M 2ε = ĨE t 1 [ I (rt B) Q 2]. 6
8 Asymptotic Optimality in Variance Reduction Theorem: M 2ε (P ε ) 2 for small ε (spatial scale). Thus, the importance sampling is optimal (or efficient). Proof: by means of Cramer s theorem for 1-dim case. For high-dimensional first passage time problem, see H. (09). 7
9 Trajectories under different measures Single Name Case 8
10 Some Modifications: SV model and Jump-Diffusion Model SV Model: JD Model: ds t S t ds t = µ S t dt + σ t S t dw t σ t = exp(y t /2) dy t = (m Y t ) dt + β dz t = µdt + σdw t + d N t j=1 (Y (j) 1), 9
11 1-dim. Default Probability - SV Model B BMC Importance Sampling (0.0010) (1.6964E-004) (0.0022) (7.3140E-004) The number of simulations is 10 4 and the Euler discretization takes time step size T/100, where T is one day. Other parameters are S 0 = 100, µ = 0.3, m = 2, α = 5, β = 1, ρ = 0. Standard errors are shown in parenthesis. 10
12 Default Probability - Jump-Diffusion Model B P JD P JD P JD P JD True Basic MC IS-JD IS-D (0.0069) (0.0024) ( ) ( ) ( ) ( ) ( ) (1 10 4) ( ) The number of simulations is 10 4 and the Euler discretization takes time step size T/100, where T is one day. Other parameters are µ = 0.06, σ = 0.2, λ = 1, a = 0, b 2 = 0.02, T = 1/252. Standard errors are shown in parenthesis. 11
13 A Nonparametric Method to Estimate Vol. Fourier Transform Method Assume a difussion process du(t) = µ(t)dt + σ(t)dw t, Task: to estimate is σ(t), i.e. the time series volatility. Malliavin and Mancino(2002,2005,2009) 12
14 Fourier Transform Method(Step 1) Compute the Fourier coefficients of du by Then, u(t) = a 0 + a 0 (du) = 1 2π a k (du) = 1 π b k (du) = 1 π k=1 [ 2π 0 2π 0 2π 0 b k(du) k du(t), cos(kt)du(t), sin(kt)du(t). cos(kt) + a k(du) k 13 sin(kt) ].
15 Fourier Transform Method(Step 2) Fourier coefficients of variance σ 2, a 0 (σ 2 ) = lim N a k (σ 2 ) = lim N b k (σ 2 ) = lim N π N [ a 2 N + 1 n s (du) + b 2 s (du)], 0 s=n 0 2π N [a s (du)a s+k (du)], k > 0, N + 1 n 0 s=n 0 2π N [a s (du)b s+k (du)], k 0, N + 1 n 0 s=n 0 where n 0 is any positive integer. so that σ 2 N (t) = N k=0 [ ak (σ 2 ) cos(kt) + b k (σ 2 ) sin(kt) ]. 14
16 Fourier Transform Method(Step 3) Reconstruct the time series variance σ 2 (t). Finally, σ 2 N (t) is an approximation of σ2 (t) as N approaches infinity, which can be given by classical Fourier-Fejer inversion formula. σ 2 (t) = lim N σ2 N (t) in prob. 15
17 Smoothing We add a function into the final computation of time series variance in order to smooth it. σ 2 (t) = lim N N ϕ (δk) [ a k (σ 2 ) cos(kt) + b k (σ 2 ) sin(kt) ] k=0 where ϕ(x) = sin2 (x) x 2 is a function in order to smooth the trajectory and δ is a smoothing parameter. 16
18 Boundary Effect Removed Simulated Data 17
19 A Price Correction Scheme: First Order Idea: (Nonlinear) Least Squares Method for first-order correction r t σ t δ t ɛ t exp( ( a + b Ŷ t ) /2) δt ɛ t. Then by MLE to regress out a and b ln r2 t δ 2 t = a + b Ŷ t + ln ɛ 2 t. 18
20 Stability of parameters - GBP/USD 19
21 Stability of parameters - JPY/USD 20
22 Back Testing Empirical LRcc Test Data Sample Period: 1993/1/5 2009/7/24 1% VaR RiskMetrics H.S. SV Model AUD O X O JPY X X O SGD X X X CAD X X O KRW X X X GBP X X O 21
23 Conclusion Simple and efficient importance sampling methods are proposed, justified by large deviation theory. Remove boundary effect of Fourier Transform Method Some empirical studies on FX data 22
24 Acknowledgments Math. Inst., Academia Sinica, Anita Chang (QF, NTHU), Tzu-Ying Chen (QF, NTHU).
25 Thank You 23
Stochastic Computation in Finance
Stochastic Computation in Finance Chuan-Hsiang Han Dept. of Quantitative Finance, NTHU Dept of Math & CS Education TMUE November 3, 2008 Outline History of Math and Finance: Fundamental Problems in Modern
More informationThe Use of Importance Sampling to Speed Up Stochastic Volatility Simulations
The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations.
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationOption pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard
Option pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard Indifference pricing and the minimal entropy martingale measure Fred Espen Benth Centre of Mathematics for Applications
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (42 pts) Answer briefly the following questions. 1. Questions
More informationarxiv: v2 [q-fin.pr] 23 Nov 2017
VALUATION OF EQUITY WARRANTS FOR UNCERTAIN FINANCIAL MARKET FOAD SHOKROLLAHI arxiv:17118356v2 [q-finpr] 23 Nov 217 Department of Mathematics and Statistics, University of Vaasa, PO Box 7, FIN-6511 Vaasa,
More informationRohini Kumar. Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque)
Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days,
More informationAsymptotic Method for Singularity in Path-Dependent Option Pricing
Asymptotic Method for Singularity in Path-Dependent Option Pricing Sang-Hyeon Park, Jeong-Hoon Kim Dept. Math. Yonsei University June 2010 Singularity in Path-Dependent June 2010 Option Pricing 1 / 21
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Basic Concepts and Techniques of Risk Management Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationEstimating Joint Default Probability by Efficient Importance Sampling with Applications from Bottom Up
Estimating Joint Default Probability by Efficient Importance Sampling with Applications from Bottom Up Chuan-Hsiang Han October 6, 2009 Abstract his paper provides a unified approach to estimate the probability
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More information"Pricing Exotic Options using Strong Convergence Properties
Fourth Oxford / Princeton Workshop on Financial Mathematics "Pricing Exotic Options using Strong Convergence Properties Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz Prof. Mike
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationOPTIONS pricing has been being a topic in the field of
IAENG International Journal of Applied Mathematics, 45:1, IJAM_45_1_7 A Simple Control Variate Method for Options Pricing with Stochastic Volatility Models Guo Liu, Qiang Zhao, and Guiding Gu Astract In
More informationNumerical schemes for SDEs
Lecture 5 Numerical schemes for SDEs Lecture Notes by Jan Palczewski Computational Finance p. 1 A Stochastic Differential Equation (SDE) is an object of the following type dx t = a(t,x t )dt + b(t,x t
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Vienna, 16 November 2007 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November
More informationNumerical Solution of Stochastic Differential Equations with Jumps in Finance
Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen School of Finance and Economics and School of Mathematical Sciences University of Technology, Sydney Kloeden,
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
More informationA No-Arbitrage Theorem for Uncertain Stock Model
Fuzzy Optim Decis Making manuscript No (will be inserted by the editor) A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao Received: date / Accepted: date Abstract Stock model is used to describe
More informationPortfolio Optimization using Conditional Sharpe Ratio
International Letters of Chemistry, Physics and Astronomy Online: 2015-07-01 ISSN: 2299-3843, Vol. 53, pp 130-136 doi:10.18052/www.scipress.com/ilcpa.53.130 2015 SciPress Ltd., Switzerland Portfolio Optimization
More informationGreek parameters of nonlinear Black-Scholes equation
International Journal of Mathematics and Soft Computing Vol.5, No.2 (2015), 69-74. ISSN Print : 2249-3328 ISSN Online: 2319-5215 Greek parameters of nonlinear Black-Scholes equation Purity J. Kiptum 1,
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationThe data-driven COS method
The data-driven COS method Á. Leitao, C. W. Oosterlee, L. Ortiz-Gracia and S. M. Bohte Delft University of Technology - Centrum Wiskunde & Informatica Reading group, March 13, 2017 Reading group, March
More informationLocal vs Non-local Forward Equations for Option Pricing
Local vs Non-local Forward Equations for Option Pricing Rama Cont Yu Gu Abstract When the underlying asset is a continuous martingale, call option prices solve the Dupire equation, a forward parabolic
More informationRisk Measurement in Credit Portfolio Models
9 th DGVFM Scientific Day 30 April 2010 1 Risk Measurement in Credit Portfolio Models 9 th DGVFM Scientific Day 30 April 2010 9 th DGVFM Scientific Day 30 April 2010 2 Quantitative Risk Management Profit
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More informationVariance Reduction for Monte Carlo Simulation in a Stochastic Volatility Environment
Variance Reduction for Monte Carlo Simulation in a Stochastic Volatility Environment Jean-Pierre Fouque Tracey Andrew Tullie December 11, 21 Abstract We propose a variance reduction method for Monte Carlo
More informationRisk Minimization Control for Beating the Market Strategies
Risk Minimization Control for Beating the Market Strategies Jan Večeř, Columbia University, Department of Statistics, Mingxin Xu, Carnegie Mellon University, Department of Mathematical Sciences, Olympia
More informationMathematics in Finance
Mathematics in Finance Steven E. Shreve Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA shreve@andrew.cmu.edu A Talk in the Series Probability in Science and Industry
More informationHedging with Life and General Insurance Products
Hedging with Life and General Insurance Products June 2016 2 Hedging with Life and General Insurance Products Jungmin Choi Department of Mathematics East Carolina University Abstract In this study, a hybrid
More informationSupplementary Appendix to The Risk Premia Embedded in Index Options
Supplementary Appendix to The Risk Premia Embedded in Index Options Torben G. Andersen Nicola Fusari Viktor Todorov December 214 Contents A The Non-Linear Factor Structure of Option Surfaces 2 B Additional
More informationCalculating Implied Volatility
Statistical Laboratory University of Cambridge University of Cambridge Mathematics and Big Data Showcase 20 April 2016 How much is an option worth? A call option is the right, but not the obligation, to
More informationThe data-driven COS method
The data-driven COS method Á. Leitao, C. W. Oosterlee, L. Ortiz-Gracia and S. M. Bohte Delft University of Technology - Centrum Wiskunde & Informatica CMMSE 2017, July 6, 2017 Álvaro Leitao (CWI & TUDelft)
More informationMultiscale Stochastic Volatility Models
Multiscale Stochastic Volatility Models Jean-Pierre Fouque University of California Santa Barbara 6th World Congress of the Bachelier Finance Society Toronto, June 25, 2010 Multiscale Stochastic Volatility
More informationEFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS
Commun. Korean Math. Soc. 23 (2008), No. 2, pp. 285 294 EFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS Kyoung-Sook Moon Reprinted from the Communications of the Korean Mathematical Society
More informationTime-changed Brownian motion and option pricing
Time-changed Brownian motion and option pricing Peter Hieber Chair of Mathematical Finance, TU Munich 6th AMaMeF Warsaw, June 13th 2013 Partially joint with Marcos Escobar (RU Toronto), Matthias Scherer
More informationLogarithmic derivatives of densities for jump processes
Logarithmic derivatives of densities for jump processes Atsushi AKEUCHI Osaka City University (JAPAN) June 3, 29 City University of Hong Kong Workshop on Stochastic Analysis and Finance (June 29 - July
More informationSolving the Black-Scholes Equation
Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Initial Value Problem for the European Call rf = F t + rsf S + 1 2 σ2 S 2 F SS for (S,
More informationStatistical Methods in Financial Risk Management
Statistical Methods in Financial Risk Management Lecture 1: Mapping Risks to Risk Factors Alexander J. McNeil Maxwell Institute of Mathematical Sciences Heriot-Watt University Edinburgh 2nd Workshop on
More informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
More information- 1 - **** d(lns) = (µ (1/2)σ 2 )dt + σdw t
- 1 - **** These answers indicate the solutions to the 2014 exam questions. Obviously you should plot graphs where I have simply described the key features. It is important when plotting graphs to label
More informationOptimal Stopping for American Type Options
Optimal Stopping for Department of Mathematics Stockholm University Sweden E-mail: silvestrov@math.su.se ISI 2011, Dublin, 21-26 August 2011 Outline of communication Multivariate Modulated Markov price
More informationLinearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing
Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing Liuren Wu, Baruch College Joint work with Peter Carr and Xavier Gabaix at New York University Board of
More informationConvergence Analysis of Monte Carlo Calibration of Financial Market Models
Analysis of Monte Carlo Calibration of Financial Market Models Christoph Käbe Universität Trier Workshop on PDE Constrained Optimization of Certain and Uncertain Processes June 03, 2009 Monte Carlo Calibration
More informationErrata, Mahler Study Aids for Exam 3/M, Spring 2010 HCM, 1/26/13 Page 1
Errata, Mahler Study Aids for Exam 3/M, Spring 2010 HCM, 1/26/13 Page 1 1B, p. 72: (60%)(0.39) + (40%)(0.75) = 0.534. 1D, page 131, solution to the first Exercise: 2.5 2.5 λ(t) dt = 3t 2 dt 2 2 = t 3 ]
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2011, Mr. Ruey S. Tsay. Solutions to Final Exam.
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2011, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (32 pts) Answer briefly the following questions. 1. Suppose
More informationGraduate School of Business, University of Chicago Business 41202, Spring Quarter 2007, Mr. Ruey S. Tsay. Solutions to Final Exam
Graduate School of Business, University of Chicago Business 41202, Spring Quarter 2007, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (30 pts) Answer briefly the following questions. 1. Suppose that
More informationPricing Variance Swaps under Stochastic Volatility Model with Regime Switching - Discrete Observations Case
Pricing Variance Swaps under Stochastic Volatility Model with Regime Switching - Discrete Observations Case Guang-Hua Lian Collaboration with Robert Elliott University of Adelaide Feb. 2, 2011 Robert Elliott,
More informationContinuous Time Finance. Tomas Björk
Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying
More informationBlack-Scholes Option Pricing
Black-Scholes Option Pricing The pricing kernel furnishes an alternate derivation of the Black-Scholes formula for the price of a call option. Arbitrage is again the foundation for the theory. 1 Risk-Free
More informationComprehensive Exam. August 19, 2013
Comprehensive Exam August 19, 2013 You have a total of 180 minutes to complete the exam. If a question seems ambiguous, state why, sharpen it up and answer the sharpened-up question. Good luck! 1 1 Menu
More informationLIBOR models, multi-curve extensions, and the pricing of callable structured derivatives
Weierstrass Institute for Applied Analysis and Stochastics LIBOR models, multi-curve extensions, and the pricing of callable structured derivatives John Schoenmakers 9th Summer School in Mathematical Finance
More informationThe Black-Scholes Equation using Heat Equation
The Black-Scholes Equation using Heat Equation Peter Cassar May 0, 05 Assumptions of the Black-Scholes Model We have a risk free asset given by the price process, dbt = rbt The asset price follows a geometric
More informationNumerical Solution of Stochastic Differential Equations with Jumps in Finance
Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen School of Finance and Economics and School of Mathematical Sciences University of Technology, Sydney Kloeden,
More information2.1 Mathematical Basis: Risk-Neutral Pricing
Chapter Monte-Carlo Simulation.1 Mathematical Basis: Risk-Neutral Pricing Suppose that F T is the payoff at T for a European-type derivative f. Then the price at times t before T is given by f t = e r(t
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationIMPA Commodities Course : Forward Price Models
IMPA Commodities Course : Forward Price Models Sebastian Jaimungal sebastian.jaimungal@utoronto.ca Department of Statistics and Mathematical Finance Program, University of Toronto, Toronto, Canada http://www.utstat.utoronto.ca/sjaimung
More informationLévy models in finance
Lévy models in finance Ernesto Mordecki Universidad de la República, Montevideo, Uruguay PASI - Guanajuato - June 2010 Summary General aim: describe jummp modelling in finace through some relevant issues.
More informationAnalytical formulas for local volatility model with stochastic. Mohammed Miri
Analytical formulas for local volatility model with stochastic rates Mohammed Miri Joint work with Eric Benhamou (Pricing Partners) and Emmanuel Gobet (Ecole Polytechnique Modeling and Managing Financial
More informationOn modelling of electricity spot price
, Rüdiger Kiesel and Fred Espen Benth Institute of Energy Trading and Financial Services University of Duisburg-Essen Centre of Mathematics for Applications, University of Oslo 25. August 2010 Introduction
More informationPricing in markets modeled by general processes with independent increments
Pricing in markets modeled by general processes with independent increments Tom Hurd Financial Mathematics at McMaster www.phimac.org Thanks to Tahir Choulli and Shui Feng Financial Mathematics Seminar
More informationAccelerated Variance Reduction Methods on GPU
Accelerated Variance Reduction Methods on GPU Chuan-Hsiang Han Department of Quantitative Finance, ational Tsing-Hua University, Hsinchu, Taiwan, R.O.C. chhan@mx.nthu.edu.tw Yu-Tuan Lin Institute of Mathematics
More informationAsymptotic methods in risk management. Advances in Financial Mathematics
Asymptotic methods in risk management Peter Tankov Based on joint work with A. Gulisashvili Advances in Financial Mathematics Paris, January 7 10, 2014 Peter Tankov (Université Paris Diderot) Asymptotic
More informationOn the pricing equations in local / stochastic volatility models
On the pricing equations in local / stochastic volatility models Hao Xing Fields Institute/Boston University joint work with Erhan Bayraktar, University of Michigan Kostas Kardaras, Boston University Probability
More informationMonte Carlo Simulation of a Two-Factor Stochastic Volatility Model
Monte Carlo Simulation of a Two-Factor Stochastic Volatility Model asymptotic approximation formula for the vanilla European call option price. A class of multi-factor volatility models has been introduced
More informationValuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model
Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 1(23) Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Generating Random Variables and Stochastic Processes Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationMonte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I January
More informationPath Dependent British Options
Path Dependent British Options Kristoffer J Glover (Joint work with G. Peskir and F. Samee) School of Finance and Economics University of Technology, Sydney 18th August 2009 (PDE & Mathematical Finance
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationIntroduction Taylor s Theorem Einstein s Theory Bachelier s Probability Law Brownian Motion Itô s Calculus. Itô s Calculus.
Itô s Calculus Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October 21, 2016 Christopher Ting QF 101 Week 10 October
More informationTheoretical Problems in Credit Portfolio Modeling 2
Theoretical Problems in Credit Portfolio Modeling 2 David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Jiaotong University(SJTU) November 3, 2017 Presented at the University of South California
More informationChange of Measure (Cameron-Martin-Girsanov Theorem)
Change of Measure Cameron-Martin-Girsanov Theorem Radon-Nikodym derivative: Taking again our intuition from the discrete world, we know that, in the context of option pricing, we need to price the claim
More informationValuation of Volatility Derivatives. Jim Gatheral Global Derivatives & Risk Management 2005 Paris May 24, 2005
Valuation of Volatility Derivatives Jim Gatheral Global Derivatives & Risk Management 005 Paris May 4, 005 he opinions expressed in this presentation are those of the author alone, and do not necessarily
More informationRisk Reward Optimisation for Long-Run Investors: an Empirical Analysis
GoBack Risk Reward Optimisation for Long-Run Investors: an Empirical Analysis M. Gilli University of Geneva and Swiss Finance Institute E. Schumann University of Geneva AFIR / LIFE Colloquium 2009 München,
More informationSolving the Black-Scholes Equation
Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Initial Value Problem for the European Call The main objective of this lesson is solving
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationExam Quantitative Finance (35V5A1)
Exam Quantitative Finance (35V5A1) Part I: Discrete-time finance Exercise 1 (20 points) a. Provide the definition of the pricing kernel k q. Relate this pricing kernel to the set of discount factors D
More informationStructural Models of Credit Risk and Some Applications
Structural Models of Credit Risk and Some Applications Albert Cohen Actuarial Science Program Department of Mathematics Department of Statistics and Probability albert@math.msu.edu August 29, 2018 Outline
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationA GENERAL FORMULA FOR OPTION PRICES IN A STOCHASTIC VOLATILITY MODEL. Stephen Chin and Daniel Dufresne. Centre for Actuarial Studies
A GENERAL FORMULA FOR OPTION PRICES IN A STOCHASTIC VOLATILITY MODEL Stephen Chin and Daniel Dufresne Centre for Actuarial Studies University of Melbourne Paper: http://mercury.ecom.unimelb.edu.au/site/actwww/wps2009/no181.pdf
More informationMath 416/516: Stochastic Simulation
Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationFinancial Mathematics and Supercomputing
GPU acceleration in early-exercise option valuation Álvaro Leitao and Cornelis W. Oosterlee Financial Mathematics and Supercomputing A Coruña - September 26, 2018 Á. Leitao & Kees Oosterlee SGBM on GPU
More informationAsymmetric information in trading against disorderly liquidation of a large position.
Asymmetric information in trading against disorderly liquidation of a large position. Caroline Hillairet 1 Cody Hyndman 2 Ying Jiao 3 Renjie Wang 2 1 ENSAE ParisTech Crest, France 2 Concordia University,
More informationMachine Learning for Quantitative Finance
Machine Learning for Quantitative Finance Fast derivative pricing Sofie Reyners Joint work with Jan De Spiegeleer, Dilip Madan and Wim Schoutens Derivative pricing is time-consuming... Vanilla option pricing
More informationOn Using Shadow Prices in Portfolio optimization with Transaction Costs
On Using Shadow Prices in Portfolio optimization with Transaction Costs Johannes Muhle-Karbe Universität Wien Joint work with Jan Kallsen Universidad de Murcia 12.03.2010 Outline The Merton problem The
More informationNumerical Methods for Pricing Energy Derivatives, including Swing Options, in the Presence of Jumps
Numerical Methods for Pricing Energy Derivatives, including Swing Options, in the Presence of Jumps, Senior Quantitative Analyst Motivation: Swing Options An electricity or gas SUPPLIER needs to be capable,
More informationMSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013
MSc Financial Engineering 2012-13 CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL To be handed in by monday January 28, 2013 Department EMS, Birkbeck Introduction The assignment consists of Reading
More informationHomework Assignments
Homework Assignments Week 1 (p. 57) #4.1, 4., 4.3 Week (pp 58 6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15 19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9 31) #.,.6,.9 Week 4 (pp 36 37)
More informationθ(t ) = T f(0, T ) + σ2 T
1 Derivatives Pricing and Financial Modelling Andrew Cairns: room M3.08 E-mail: A.Cairns@ma.hw.ac.uk Tutorial 10 1. (Ho-Lee) Let X(T ) = T 0 W t dt. (a) What is the distribution of X(T )? (b) Find E[exp(
More informationA Note about the Black-Scholes Option Pricing Model under Time-Varying Conditions Yi-rong YING and Meng-meng BAI
2017 2nd International Conference on Advances in Management Engineering and Information Technology (AMEIT 2017) ISBN: 978-1-60595-457-8 A Note about the Black-Scholes Option Pricing Model under Time-Varying
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationAsset Pricing Models with Underlying Time-varying Lévy Processes
Asset Pricing Models with Underlying Time-varying Lévy Processes Stochastics & Computational Finance 2015 Xuecan CUI Jang SCHILTZ University of Luxembourg July 9, 2015 Xuecan CUI, Jang SCHILTZ University
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationStochastic modelling of electricity markets Pricing Forwards and Swaps
Stochastic modelling of electricity markets Pricing Forwards and Swaps Jhonny Gonzalez School of Mathematics The University of Manchester Magical books project August 23, 2012 Clip for this slide Pricing
More informationStochastic Processes and Brownian Motion
A stochastic process Stochastic Processes X = { X(t) } Stochastic Processes and Brownian Motion is a time series of random variables. X(t) (or X t ) is a random variable for each time t and is usually
More informationNear-Expiry Asymptotics of the Implied Volatility in Local and Stochastic Volatility Models
Mathematical Finance Colloquium, USC September 27, 2013 Near-Expiry Asymptotics of the Implied Volatility in Local and Stochastic Volatility Models Elton P. Hsu Northwestern University (Based on a joint
More information